\(\begin{array}{l}
A = \dfrac{{\cos 2x}}{{\sin x + \cos x}} \\= \dfrac{{{{\cos }^2}x - {{\sin }^2}x}}{{\sin x + \cos x}}\\
= \dfrac{{(\cos x - \sin x)(\cos x + \sin x)}}{{\sin x + \cos x}}\\
= \cos x - \sin x\\
= \sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right)
\end{array}\)
\(\begin{array}{l}
B = \dfrac{{\sin x + \tan x}}{{\tan x}} - \sin x.\cos x\\
= \dfrac{{\sin x + \frac{{\sin x}}{{\cos x}}}}{{\frac{{\sin x}}{{\cos x}}}} - \sin x.\cos x\\
= \dfrac{{\cos x\sin x + \sin x}}{{\cos x}}\dfrac{{\cos x}}{{\sin x}} - \sin x.\cos x\\
= \cos x + 1 - \sin x.\cos x\\
\\
C = \tan x + \dfrac{{\cos x}}{{1 + \sin x}}\\
= \dfrac{{\sin x}}{{\cos x}} + \frac{{\cos x}}{{1 + \sin x}}\\
= \dfrac{{\sin x + {{\sin }^2}x + {{\cos }^2}x}}{{\cos x(1 + \sin x)}}\\
= \dfrac{{1 + \sin x}}{{\cos x(1 + \sin x)}}\\
= \dfrac{1}{{\cos x}}
\end{array}\)
